Incompleteness: The Proof and Paradox of Kurt Gödel
Gödel was a hacker. He attended the meetings of the Vienna Circle, one
of the most important philosophical groups in history, and sat quietly,
convinced they were all wrong. Others might try to argue them out of it.
Not Gödel. Perhaps (rightly) he thought that rational argument would get
him nowhere with people so detached from reality. So he decided toprovethem wrong.
The result was Gödel’s incompleteness theorem, one of the most
celebrated results in mathematics and logic, elegantly proving that any
mathematical system complicated enough to do basic arithmetic contains
statements which, while true, cannot be proven within the system.
But even this was to no avail, the Vienna Circle continued to insist the
proof was bogus or irrelevant. (The Vienna Circle insisted the world was
simply language-games, with rules and structures. Gödel, by showing
something that was clearly true to us but not provable from within the
game, thought he had proved that such things, like numbers, have an
independent reality.) Gödel, an incredibly odd and delusional figure,
remained quiet for many years, practically confiding only in Albert
Einstein, and little at all after Einstein’s death.
This is a very bad book. It rambles and prattles and occasionally
repeats itself practically verbatim. (It is the result of a project to
improve science writing simply by paying famous authors to write about
scientific topics. Perhaps the payment should be contingent on some
measure of quality in the result.) But it is a compelling story and
Gödel’s proof is so brilliantly beautiful that it should be learned by
all educated people. I had never seen it presented in any real detail
before and once I got to its key principle I exclaimed and tossed the
book down and paced, admiring its brilliance. But there are many sources
who will explain the fairly simple idea. I’d love to tell you about it
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February 12, 2007